Probabilities

We write P(A) as “the fraction of possible worlds in which A is true”.

P(A) = Area of the green oval.

Interpreting the axioms

Axioms:

• 0<=P(A)<=1.

• P(True) = 1

• P(False) = 0

• P(A or B)=P(A) + P(B) - P(A and B)

Theorems from the Axioms

0<=P(A)<=1, P(True)=1, P(False)=0.

P(A or B) = P(A) + P(B) - P(A and B).

From these we can prove:

• P(not A) = P(~A) = 1-P(A)

0<= P(A) <=1, P(True)=1, P(False)=0.

P(A or B) =P(A) + P(B) - P(A and B)

From these we can prove:

• P(A) = P(A ^ B) + P(A ^ ~B)